On a Bernoulli problem with geometric constraints

Antoine Laurain; Yannick Privat

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 1, page 157-180
  • ISSN: 1292-8119

Abstract

top
A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by x1 ≥ 0 and its boundary to contain a segment of the hyperplane  {x1 = 0}  where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed and this paper is devoted first to the study of geometric and asymptotic properties of the solution and then to the numerical treatment of the problem using a shape optimization formulation. The major difficulty and originality of this paper lies in the treatment of the geometric constraints.

How to cite

top

Laurain, Antoine, and Privat, Yannick. "On a Bernoulli problem with geometric constraints." ESAIM: Control, Optimisation and Calculus of Variations 18.1 (2012): 157-180. <http://eudml.org/doc/221908>.

@article{Laurain2012,
abstract = {A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by x1 ≥ 0 and its boundary to contain a segment of the hyperplane  \{x1 = 0\}  where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed and this paper is devoted first to the study of geometric and asymptotic properties of the solution and then to the numerical treatment of the problem using a shape optimization formulation. The major difficulty and originality of this paper lies in the treatment of the geometric constraints. },
author = {Laurain, Antoine, Privat, Yannick},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Free boundary problem; Bernoulli condition; shape optimization; free boundary problem},
language = {eng},
month = {2},
number = {1},
pages = {157-180},
publisher = {EDP Sciences},
title = {On a Bernoulli problem with geometric constraints},
url = {http://eudml.org/doc/221908},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Laurain, Antoine
AU - Privat, Yannick
TI - On a Bernoulli problem with geometric constraints
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/2//
PB - EDP Sciences
VL - 18
IS - 1
SP - 157
EP - 180
AB - A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by x1 ≥ 0 and its boundary to contain a segment of the hyperplane  {x1 = 0}  where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed and this paper is devoted first to the study of geometric and asymptotic properties of the solution and then to the numerical treatment of the problem using a shape optimization formulation. The major difficulty and originality of this paper lies in the treatment of the geometric constraints.
LA - eng
KW - Free boundary problem; Bernoulli condition; shape optimization; free boundary problem
UR - http://eudml.org/doc/221908
ER -

References

top
  1. A. Acker, An extremal problem involving current flow through distributed resistance. SIAM J. Math. Anal.12 (1981) 169–172.  Zbl0456.49007
  2. C. Atkinson and C.R. Champion, Some boundary-value problems for the equation ∇·(|∇ϕ| N∇ϕ) = 0. Quart. J. Mech. Appl. Math.37 (1984) 401–419.  Zbl0567.73054
  3. A. Beurling, On free boundary problems for the Laplace equation, Seminars on analytic functions1. Institute for Advanced Studies, Princeton (1957).  
  4. F. Bouchon, S. Clain and R. Touzani, Numerical solution of the free boundary Bernoulli problem using a level set formulation. Comput. Methods Appl. Mech. Eng.194 (2005) 3934–3948.  Zbl1090.76048
  5. E.N. Dancer and D. Daners, Domain perturbation for elliptic equations subject to Robin boundary conditions. J. Differ. Equ.138 (1997) 86–132.  Zbl0886.35063
  6. M.C. Delfour and J.-P. Zolésio, Shapes and geometries – Analysis, differential calculus, and optimization, Advances in Design and Control4. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001).  
  7. L.C. Evans, Partial differential equations, Graduate Studies in Mathematics19. American Mathematical Society, Providence (1998).  Zbl0902.35002
  8. A. Fasano, Some free boundary problems with industrial applications, in Shape optimization and free boundaries (Montreal, PQ, 1990), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.380, Kluwer Acad. Publ., Dordrecht (1992) 113–142.  Zbl0765.76005
  9. M. Flucher and M. Rumpf, Bernoulli’s free boundary problem, qualitative theory and numerical approximation. J. Reine Angew. Math.486 (1997) 165–204.  Zbl0909.35154
  10. A. Friedman, Free boundary problem in fluid dynamics, in Variational methods for equilibrium problems of fluids, Trento 1983, Astérisque118 (1984) 55–67.  
  11. A. Friedman, Free boundary problems in science and technology. Notices Amer. Math. Soc.47 (2000) 854–861.  Zbl1040.35145
  12. P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics24. Pitman (Advanced Publishing Program), Boston (1985).  Zbl0695.35060
  13. J. Haslinger, T. Kozubek, K. Kunisch and G. Peichl, Shape optimization and fictitious domain approach for solving free boundary problems of Bernoulli type. Comp. Optim. Appl.26 (2003) 231–251.  Zbl1077.49030
  14. J. Haslinger, K. Ito, T. Kozubek, K. Kunisch and G. Peichl, On the shape derivative for problems of Bernoulli type. Interfaces in Free Boundaries11 (2009) 317–330.  Zbl1178.49055
  15. A. Henrot and M. Pierre, Variation et optimisation de formes – Une analyse géométrique, Mathématiques & Applications48. Springer, Berlin (2005).  Zbl1098.49001
  16. A. Henrot and H. Shahgholian, Existence of classical solutions to a free boundary problem for the p-Laplace operator. I. The exterior convex case. J. Reine Angew. Math.521 (2000) 85–97.  Zbl0955.35078
  17. A. Henrot and H. Shahgholian, Existence of classical solutions to a free boundary problem for the p-Laplace operator. II. The interior convex case. Indiana Univ. Math. J.49 (2000) 311–323.  Zbl0977.35148
  18. A. Henrot and H. Shahgholian, The one phase free boundary problem for the p-Laplacian with non-constant Bernoulli boundary condition. Trans. Amer. Math. Soc.354 (2002) 2399–2416.  Zbl0988.35174
  19. K. Ito, K. Kunisch and G.H. Peichl, Variational approach to shape derivatives for a class of Bernoulli problems. J. Math. Anal. Appl.314 (2006) 126–149.  Zbl1088.49028
  20. C.T. Kelley, Iterative methods for optimization, Frontiers in Applied Mathematics18. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1999).  Zbl0934.90082
  21. V.A. Kondratév, Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obšč.16 (1967) 209–292.  
  22. C.M. Kuster, P.A. Gremaud and R. Touzani, Fast numerical methods for Bernoulli free boundary problems. SIAM J. Sci. Comput.29 (2007) 622–634.  Zbl1136.65113
  23. J. Lamboley and A. Novruzi, Polygons as optimal shapes with convexity constraint. SIAM J. Control Optim.48 (2009) 3003–3025.  Zbl1202.49053
  24. E. Lindgren and Y. Privat, A free boundary problem for the Laplacian with a constant Bernoulli-type boundary condition. Nonlinear Anal.67 (2007) 2497–2505.  Zbl1123.35092
  25. J. Nocedal and S.J. Wright, Numerical optimization. Springer Series in Operations Research and Financial Engineering, Springer, New York, 2nd edition (2006).  Zbl1104.65059
  26. J.R. Philip, n-diffusion. Austral. J. Phys.14 (1961) 1–13.  
  27. J. Sokołowski and J.-P. Zolésio, Introduction to shape optimization : Shape sensitivity analysis, Springer Series in Computational Mathematics16. Springer-Verlag, Berlin (1992).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.